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This paper is made available online in accordance with publisher policies. Please scroll down to view the document itself. Please refer to the repository record for this item and our policy information available from the repository home page for further information. To see the final version of this paper please visit the publisher’s website. Access to the published version may require a subscription. Author(s): J. Southworth, J. V. Clausen Article Title: Absolute dimensions of eclipsing binaries - XXIV. The Be star system DW Carinae, a member of the open cluster Collinder 228 Year of publication: 2007 Link to published article: http://dx.doi.org/10.1051/0004-6361:20065614 Publisher statement: © ESO 2007. Southworth, J. and Clausen, J. V. (2007). Absolute dimensions of eclipsing binaries - XXIV. The Be star system DW Carinae, a member of the open cluster Collinder 228. Astronomy & Astrophysics, Vol.461 (3), pp.1077-1093

A&A 461, 1077–1093 (2007) Astronomy DOI: 10.1051/0004-6361:20065614 & c ESO 2007 Astrophysics

Absolute dimensions of eclipsing binaries

XXIV. The Be star system DW Carinae, a member of the open cluster Collinder 228,

J. Southworth1,2 and J. V. Clausen1

1 Niels Bohr Institute, Copenhagen University, Juliane Maries Vej 30, 2100 Copenhagen Ø, Denmark 2 Department of Physics, University of Warwick, Coventry, CV4 7AL, UK e-mail: [email protected] (JS), [email protected] (JVC)

Received 16 May 2006 / Accepted 11 October 2006

ABSTRACT

Context. The study of detached eclipsing binaries which are members of stellar clusters is is a powerful way of determining the properties of the cluster and of constraining the physical ingredients of theoretical stellar evolutionary models. Aims. DW Carinae is a close but detached early B-type eclipsing binary in the young open cluster Collinder 228. We have measured accurate physical properties of the components of DW Car (masses and radii to 1%, effective temperatures to 0.02 dex) and used these to derive the age, metallicity and distance of Collinder 228. Methods. The rotational velocities of both components of DW Car are high, so we have investigated the performance of double- Gaussian fitting, one- and two-dimensional cross-correlation and spectral disentangling for deriving spectroscopic radial velocites in the presence of strong line blending. Gaussian and cross-correlation analyses require substantial corrections for the effects of line blending, which are only partially successful for cross-correlation. Spectral disentangling is to be preferred because it does not assume anything about the shapes of spectral lines, and is not significantly affected by blending. However, it suffers from a proliferation of local minima in the least-squares fit. We show that the most reliable radial velocities are obtained using spectral disentangling constrained by the results of Gaussian fitting. Complete Strömgren uvby light curves have been obtained and accurate radii have been measured from them by modelling the light curves using the Wilson-Devinney program. This procedure also suffers from the presence of many local minima in parameter space, so we have constrained the solution using an accurate spectroscopic light ratio. The effective temperatures and reddening of the system have been found from Strömgren photometric calibrations. Results. The mass and radius of DW Car A are MA = 11.34 ± 0.12 M and RA = 4.558 ± 0.045 R. The values for DW Car B are MB = 10.63 ± 0.14 M and RB = 4.297 ± 0.055 R. Strömgren photometric calibrations give effective temperatures of Teff A = 27 900 ± 1000 K and TeffB = 26 500 ± 1000 K, and a reddening of Eb−y = 0.18 ± 0.02, where the quoted uncertainties include a contribution from the intrinsic uncertainty of the calibrations. The membership of DW Car in Cr 228 allows us to measure the distance, age and chemical composition of the cluster. We have used empirical bolometric corrections to calculate a distance modulus of 12.24 ± 0.12 mag for DW Car, which is in agreement with, and more accurate than, literature values. A comparison between the properties of DW Car and the predictions of recent theoretical evolutionary models is undertaken in the mass-radius and mass-Teff diagrams. The model predictions match the measured properties of DW Car for an age of about 6 Myr and a fractional metal abundance of Z ≈ 0.01. Key words. stars: fundamental parameters – stars: binaries: eclipsing – stars: emission-line, Be – stars: distances – stars: individual: DW Carinae – open clusters and associations: individual: Collinder 228

1. Introduction radial velocity curves (e.g., Southworth et al. 2005b; Torres et al. 1997). The effective temperatures of the two stars can be found The study of detached eclipsing binary star systems (dEBs) is by spectral analysis or photometric calibrations, allowing the lu- one of the few ways in which the absolute properties of stars can minosities and distance of the stars to be calculated directly (e.g., be measured directly and accurately (Andersen 1991). These ob- Southworth et al. 2005a; Clausen & Giménez 1991). jects are therefore of fundamental importance to the understand- ing of the characteristics of single stars. The masses and radii of One of the most important uses of the physical properties of the component stars in dEBs can be determined to accuracies of dEBs is as a check of the predictions of theoretical models of better than 1% from the analysis of light curves and double-lined stellar evolution. The two components of a dEB have the same age and chemical composition, as they were born together, but in general different masses, radii and luminosities. Theoretical Based on observations carried out with the Strömgren Automatic Telescope (SAT) and the FEROS spectrograph at ESO, La Silla, Chile models must therefore be able to match their accurately-known (62.H-0319, 62.L-0284, 63.H-0080, 64.L-0031). properties for one age and chemical composition. A good match Full list of the radial velocity measurements is only available in is often achieved (e.g., Southworth et al. 2004b) for two rea- electronic form at the CDS via anonymous ftp to sons. Firstly, the models are in general fairly reliable, and the ef- cdsarc.u-strasbg.fr (130.79.128.5) or via fects of improved input physics lead to only minor adjustments http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/461/1077 in the evolutionary phases for which we are able to study dEBs.

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20065614 1078 J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae

Table 1. Identifications and astrophysical data for DW Car. T0 is a ref- The eclipsing nature of DW Car was discovered by erence time of mid-eclipse of the primary star. Hertzsprung (1924), who measured the orbital period to be 0.66382 d from a photographic light curve containing 251 ob- DW Car References servations. Van den Hoven van Genderen (1939) refined the Henry Draper Catalogue HDE 305543 1 orbital period using a 914-point photographic light curve, but Tycho number TYC 8957-1314-1 2 stated that the period should be doubled even though there was α2000 10 43 10.1 2 no clear evidence that adjacent minima had different depths. − δ2000 60 02 12 2 Gaposchkin (1952, 1953) published a photographic light curve Spectral type B1 V + B1 V 3 V 9.675 ± 0.005 4 containing over one thousand points and found an orbital period b − y 0.067 ± 0.005 4 of 1.3277504d. His estimated masses and radii are quite close to the values we measure below. m1 0.046 ± 0.008 4 c1 −0.017 ± 0.008 4 A spectroscopic orbit for DW Car was published by Ferrer β 2.528 ± 0.007 4 et al. (1985) based on 67 photographic observations. They found T0 (HJD) 2 446 828.6692(4) 4 no evidence for orbital eccentricity, and a systemic velocity Orbital period (days) 1.32774934(44) 4 which is consistent with membership of Cr 228. A spectral clas- + References: (1) Cannon (1925); (2) Høg et al. (2000); (3) Levato & sification of B1 V B1 V was provided by Levato & Malaroda Malaroda (1981); (4) Paper I; quantities in parentheses are uncertainties (1981). in the final digit of the quanities. We shall refer to the primary and secondary components of DW Car as star A and star B, respectively, where star A is eclipsed by star B at phase 0.0. Star A has a greater surface brightness and mass than star B. Secondly, there are several quantities which are not in general independently known for dEBs, including age, metal abundance and helium abundance, and these can be freely adjusted to help 1.2. Collinder 228 the models match the observed properties of a dEB. This last point means that it is extremely difficult to investigate the suc- The open cluster Cr 228 is a young and sparse cluster located cess of the treatment in theoretical models of several physical in the Carina spiral arm feature. Its immediate surroundings are phenomena, for example convective core overshooting, mixing quite nebulous, making photometric studies very difficult. length, mass loss and opacity (see Cassisi 2004, 2005). Feinstein et al. (1976) found a reddening of EB−V = 0.33 ± Extra constraints on the properties of a dEB can be obtained 0.08 mag, an age of less than 5 Myr and a distance modulus of − = ± if it is a member of a co-evolutionary group of stars (Southworth V0 MV 12.0 0.2 mag on the basis of a photoelectric UBV et al. 2004a; Hebb et al. 2004). If the age and chemical compo- study. sition of the stars is known independently then these quantities Thé et al. (1980) obtained photoelectric observations in the can be used to constrain the models, making it more challenging Walraven system and found a distance modulus of 12.0 mag as- for them to match observations and so allowing the effects of suming a normal reddening law (or 11.7 mag assuming RV = more subtle physical phenomena to be investigated (e.g., Torres AV = 4.0). EB−V & Ribas 2002). If the age and composition of the stellar group is Turner & Moffat (1980) obtained photoelectric UBV obser- not known then these may be found very precisely by comparing vations and showed that the reddening in the region was normal model predictions to the properties of the dEB (Southworth et al. and that the open clusters Trumpler 14, Trumpler 15 and Cr 228 2004a,b; Thompson et al. 2001). In addition, dEBs are excel- are at a common distance modulus of 12.15 ± 0.14 mag. They lent distance indicators (e.g., Guinan et al. 1998; Clausen 2004), found that the age of Cr 228 must be at least 1 Myr from the allowing accurate and precise distances to be found to the par- presence of main sequence stars at (B − V)0 = −0.1. ent stellar systems in several empirical and semi-empirical ways Tapia et al. (1988) obtained JHK photometry of the region, (Southworth et al. 2005a; Guinan et al. 1998). and found the reddening characteristics of Cr 228, and other nearby clusters, to be anomalous. These authors found that the ff 1.1. DW Carinae absolute visual extinction, AV , is quite di erent when calculated from the colour excess EB−V rather than from colour excesses for Here we present a study of the young, early B-type system longer-wavelength passbands. Under the assumption that EB−V DW Car, which is a member of the Southern open cluster is anomalous in this region, they found AV = 1.94 ± 0.16 mag Collinder 228 (Ferrer et al. 1985). We have obtained complete from EV−K, giving a distance modulus of V0 − MV = 11.6 ± 0.4. and accurate Strömgren uvby light curves, which are presented Carraro & Patat (2001) obtained UBVRI photometry of in Clausen et al. (2007, hereafter Paper I) along with a revised Cr 228 and several nearby clusters. They found an extinction of orbital ephemeris. We have also obtained extensive spectroscopy EB−V = 0.30 ± 0.05 mag, in agreement with Feinstein et al. using the FEROS échelle spectrograph and the 1.5 m telescope (1976) but disagreeing with Tapia et al. (1988). Their distance at ESO La Silla. modulus, 11.4 ± 0.2 mag, is smaller than that derived in most DW Car (Table 1) is composed of two similar B1 V stars in previous studies. Carraro & Patat suggested that about 30% of a 1.3 day orbit. The stars are well separated, despite the short- the stars in Cr 228 are binary, in good agreement with the spec- ness of the orbital period, as they are essentially unevolved. troscopic study of Levato et al. (1990). Such a system is particularly useful because there are very few Massey et al. (2001) obtained new spectral types for 16 stars, well-studied components of dEBs with masses of 10 M or and collected literature values for another 40 stars, in the region greater which are close to the zero-age main sequence (ZAMS) of Cr 228. They found a median EB−V of 0.37 and a distance (Andersen 1991; Gies 2003). High-mass unevolved dEBs can be modulus of 12.5 mag. They obtained the ages of the higher- particularly useful for investigating the effects of the opacities mass unevolved stars from interpolation in effective tempera- used in theoretical models (Torres et al. 1997). ture and absolute bolometric magnitude using the theoretical J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae 1079 evolutionary tracks of Schaller et al. (1992). The age and stan- Table 2. Logbook of the spectroscopic observations of DW Car. HJD dard deviation is 2.2 ± 1.0 Myr, but these authors note that refers to the exposure midpoint, ID to the identification number, SD the appearance of the HR diagram suggests a spread in ages to if it was included in the spectral disentangling analysis and texp to in Cr 228. the exposure time. The column headed “Obs” indicates the observer / It is clear from the above discussion that the physical proper- (H indicates Heidelberg Copenhagen guaranteed time observations). ties of the open cluster Cr 228 are rather uncertain, because the The signal to noise ratio, S/N, of single exposures was measured at cluster is quite sparse and located in a nebulous area. The bright- approximately 4000 and 5000 Å. est member of this cluster, QZ Car (HD 9326), is a quadruple system which contains two spectroscopic binaries, one of which HJD minus ID SD Obs Orbital texp S/N 2 400 000 phase (s) shows eclipses of a semi-detached nature (Mayer et al. 2001). 51 144.83060 s 1 y H 0.73512 600 35–60 51 145.84400 s 2 H 0.49836 600 30–55 2. Observations and data reduction 51 146.86755 s 3 y H 0.26925 600 40–75 51 147.86649 s 4 H 0.02161 600 35–60 Light curves of DW Car in the Strömgren uvby photometric sys- 51 148.80902 s 5 y H 0.73148 600 40–85 tem were obtained from the Strömgren Automated Telescope at 51 149.87170 s 6 H 0.53184 420 25–40 ESO La Silla. These are presented in Paper I and contain 518 ob- 51 150.83746 s 7 y H 0.25920 600 40–60 servations in each passband. Paper I also presents an updated 51 152.73657 s 8 H 0.68953 600 25–60 orbital ephemeris and standard Strömgren uvby and Crawford 51 172.69544 s 9 y H 0.72164 900 50–80 Hβ indices for DW Car. 51 174.78088 s 10 y H 0.29229 900 75–110 51 176.80734 s 11 y H 0.81853 600 65–95 2.1. Spectroscopic data 51 178.84953 s 12 y H 0.35661 600 70–90 51 179.87450 s 13 y H 0.12857 600 65–100 29 high-resolution spectra were obtained using the FEROS 51 180.81349 s 14 y H 0.83578 600 60–80 fibre-fed échelle spectrograph on the ESO 1.52 m telescope at 51 181.87490 s 15 y H 0.63519 600 65–90 La Silla (Kaufer et al. 1999, 2000) between 1998 November 51 182.86220 s 16 y H 0.37877 600 60–100 and 2000 January. This spectrograph is located in a temperature- 51 184.86864 s 17 H 0.88993 600 55–75 51 187.87364 s 18 H 0.15316 600 35–65 controlled room and covers the spectral region from the Balmer 51 189.86682 s 19 y H 0.65433 600 50–75 jump to 8700 Å without interruption and at a constant velocity 51 190.84867 s 20 H 0.39382 600 50–70 −1 −1 resolution of 2.7 km s px (λ/∆λ = 48 000). An observing 51 191.86279 s 21 y H 0.15760 600 50–75 log is given in Table 2. 51 192.80846 s 22 y H 0.86984 600 65–85 A modified version of the MIDAS FEROS package, prepared 51 193.87953 s 23 H 0.67652 720 65–95 by H. Hensberge, was used for the basic reduction1. This pack- 51 197.85496 s 24 y H 0.67063 600 40–65 age pays careful attention to background removal, definition and 51 199.87004 s 25 y H 0.18830 900 60–85 extraction of the individual orders and wavelength calibration, 51 209.72395 s 26 JVC 0.60981 1800 110–140 and provides a significant improvement on the standard quick- 51 382.51044 s 27 y H 0.74468 1800 50–125 51 386.50761 s 28 H 0.75517 1800 70–135 look reduction pipeline. The observations were reduced night by 51 563.86655 s 29 y JVC 0.33380 2400 105–150 night using calibration exposures (thorium-argon and flat field) obtained during the preceding afternoon and following morning for each night. Standard (rather than optimal) extraction was ap- plied, and no order merging was attempted. A standard error of 0.50 and 0.72. Whilst the emission appears to be twice as strong the wavelength calibration of about 0.002–0.003 Å was typically at phase 0.50, this is simply an artefact of the continuum normal- obtained. isation as the system is about half as bright at eclipse midpoint compared to during quadrature phases. Further investigation of 3. Radial velocity analysis the emission line strengths will require flux-calibrated spectra (D. Lennon, private communication). The FEROS spectra of DW Car cover essentially the entire op- Narrow regions around the five He I absorption lines at 4026, tical wavelength range, but are disappointingly short of stellar 4388, 4471, 4922 and 6678 Å were chosen for our radial veloc- features. The only absorption lines which are easy to discern are ity analysis. The remaining visible He I lines are all broad and the hydrogen Balmer lines and several He I lines. The high rota- very shallow, and we have found that minor errors in continuum −1 tional velocity of both stars, approximately 170 km s , twinned placement can cause a large systematic error in the measured with the fact that the spectrum of each star is diluted by the flux centres and widths of these lines. The hydrogen Balmer lines of the companion, means that only the strongest spectral lines were not included in the radial velocity analysis, as line blend- are visible. ing can cause large and systematic underestimation of the veloc- The Hα line shows significant emission with an absorption ity semiamplitudes (Petrie et al. 1967; Hilditch 1973; Andersen superimposed on the line core. The emission and emission ab- 1975). Regions from an example spectrum around the lines used sorption do not follow the orbital motion of the stars but remain are shown in Fig. 2. at the systemic velocity of the system. Slight emission is also As there are very few spectral lines strong enough to yield visible in the centre of Hβ for some of the spectra, depending on reliable radial velocities, we have used four different methods to signal to noise ratio. Figure 1 shows Hα spectra at phases 0.27, measure these velocities, allowing us to investigate which are the 1 most reliable. The techniques are fitting double Gaussian func- See these websites for further details: gauss onecor http://www.ls.eso.org/lasilla/sciops/2p2/E2p2M/FEROS/ tions (hereafter called 2 ), cross-correlation ( ), DRS two-dimensional cross-correlation (todcor) and spectral disen- http://www.ls.eso.org/lasilla/sciops/2p2/E1p5M/FEROS/ tangling. Spectroscopic orbits were calculated for each spectral Reports line and for each technique. 1080 J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae

Fig. 1. Continuum-normalised Hα profiles of DW Car at phases 0.29, 0.50 and 0.72 (spectra s 10, s 2 and s 9). Spectra s 2 and s 9 have been shifted by +1and+2, respectively, for clarity.

The onecor and todcor algorithms require a template spectrum which closely matches the spectral characteristics of the components of DW Car. We have used a FEROS spectrum Fig. 2. Continuum-normalised example spectra in the regions of the of the rotational velocity standard star HR 1855 (υ Ori, spec- − five He I lines used in the radial velocity analysis. Spectrum s 29 tral type B0 V), which has a rotational velocity of 10–20 km s 1 (phase 0.33) is shown in the top four panels and spectrum s 27 (Lyubimkov et al. 2004; Abt et al. 2002). This spectrum has sig- (phase 0.74) in the last panel. Shallow lines can also be seen at 4009 Å nal to noise ratios of between 235 and 350 depending on the (He I) and 4481 (Mg II). We have chosen spectra with a high signal to échelle order, and was broadened by a rotational velocity of noise ratio to plot – most of the spectra have a significantly lower ratio. 150 km s−1 using the method of Gray (1992) with linear limb 2 darkening coefficients of between 0.39 and 0.30 depending on implemented in the sbop program (Etzel 2004), which is the wavelength of the spectral order. We have measured the ra- a modified and expanded version of an earlier code by Wolfe dial velocity of this star to be +19.9 ± 2.4kms−1 from Gaussian et al. (1967). The ephemeris from Paper I was adopted (Table 1) fits to several He I lines, in agreement with the Wilson-Evans- and the orbit was assumed to be circular based on initial analy- Batten radial velocity catalogue (Duflot et al. 1995) value of ses, the shortness of the orbital period and the shape of the light +17.4kms−1. We have subtracted 19.9 km s−1 from all the ve- curve. locities of DW Car measured using HR 1855 as a template. The means and standard deviations of the full widths at half maximum of the fitted Gaussian functions are given in Table 3.

3.1. Gaussian fitting (2GAUSS) 3.2. One-dimensional cross-correlation (ONECOR) The spectra of DW Car were velocity-binned to 5 km s−1 px−1 and a double Gaussian function was fitted to four of the five good The velocity-binned spectra of DW Car were subjected to He I lines (listed above) by the method of least squares with re- a cross-correlation analysis (Simkin 1974; Tonry & Davis 1979) jection of points deviating by more than 4σ from an initial best using Fast Fourier Transform techniques. We investigated using ff fit. The fit to each spectrum was inspected by eye and poor fits three di erent types of template spectra to derive radial veloci- were rejected. Double weights were assigned to a few fits which ties from the target spectra: seemed to be of a particularly high quality. – We attempted to use spectrum s 2 as a template, which was Spectroscopic orbits were fitted to the measured radial ve- observed at conjunction (phase 0.498), because this provides locities for each spectral line and the systemic velocities of the two stars were not forced to be equal (see Popper & Hill 1991). 2 Spectroscopic Binary Orbit Program, The orbits were calculated using the method of Lehman-Filhés http://mintaka.sdsu.edu/faculty/etzel/ J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae 1081

Table 3. Additional parameters found during the spectroscopic anal- 3.4. Spectral disentangling ysis. The full widths at half maximum of the
Gaussian functions come from the 2gauss analysis and light ratios B come from the The spectral disentangling technique solves simultaneously for A todcor analysis. the contributions of the components to the observed compos- ite spectra, and for the Doppler shifts in the component spec-

−1 tra. We have applied the method introduced by Simon & Sturm He I line FWsHM of the Gaussian fits (km s ) Light ratio (1994) using a version of the original code revised for the (Å) Star A Star B from todcor ± ± ± Linux operating system. It assumes a constant light level, so 4026.189 171.5 9.6 161.9 11.20.839 0.138 observations taken during eclipse were rejected from the anal- 4387.928 167.8 ± 10.1 163.8 ± 6.70.855 ± 0.106 4471.681 194.9 ± 16.2 200.8 ± 21.60.893 ± 0.059 ysis; DW Car is constant to within about 5% outside eclipses. 4921.929 173.9 ± 9.2 163.8 ± 10.60.856 ± 0.082 Dedicated IDL programs were applied to remove cosmic ray events and other defects, and for careful normalization of the individual orders. the best match to the target spectra. These attempts failed We have assumed the ephemeris from Paper I and a circular because bad pixel values in the target spectra, caused by orbit, and fitted for the velocity semamplitudes of the spectro- CCD defects, tended to correlate against their couterparts in scopic orbit for each star, KA and KB. Suitable spectral lines are the template spectrum and deform the cross-correlation func- present only in orders 55, 51, 50, 45 and 33, and these orders were analysed individually. The formal errors in KA and KB re- tion (CCF). −1 – The artificially broadened spectrum of HR 1855 was used turned by the code are generally around 0.3 km s ,whichis much smaller than expected given the range in KA and KB for and found to give good results. This spectrum does not con- ff tain the bad pixels common to the spectra of DW Car. We di erent orders (Tables 4, 5). It is not clear how to calculate ro- therefore used this template to measure radial velocities. bust uncertainties for spectroscopic orbits calculated in disen- – Synthetic spectra generated using the uclsyn code (see tangling analyses (Hynes & Maxted 1998), so we will not quote Southworth et al. 2004a for references). The results were in- uncertainties for individual spectroscopic orbits. We have also disentangled the spectra for a grid of fixed val- accurate because the synthetic profiles were a poor match −1 −1 uclsyn ues of KA (from 245 to 290 km s )andKB (260 to 305 km s ) to the observations, as expected because is not in- −1 tended for stars this hot. We did not pursue the use of syn- with a spacing of 1 km s , to study how the quality of the least- thetic spectra further because good results had already been squares fit varies. The results are shown in Fig. 3 for the spec- obtained using template spectra of HR 1855. tra from échelle order 55. There are several local maxima but one clear global maximum. The highest point of a surface fitted Radial velocities were accepted or rejected by eye, based on the by least squares around the global maximum is indicated. This shape of the CCF. The peaks of the CCF were then determined diagram shows that whilst spectral disentangling can be used to by least-squares quadratic interpolation and spectroscopic orbits determine spectroscopic orbits for binary stars, the best-fitting were calculated using sbop as before (Sect. 3.1). We experi- solution returned by minimisation algorithms cannot in general mented with using unbroadened template spectra as these should be trusted unless confirmed to be the global maximum by a grid be less strongly affected by line blending, but found that the re- search or by external information. sults were excessively noisy. This effect has been noted before and is due to the template and target spectra being quite different 3.5. Corrections from synthetic spectra to each other. The need to apply corrections to the radial velocities derived using the todcor algorithm has been demonstrated in several 3.3. Two-dimensional cross-correlation (TODCOR) cases, e.g., Torres et al. (1997) and Torres & Ribas (2002). These The todcor algorithm (Zucker & Mazeh 1994) produces a two- corrections may be needed to remove small radial velocity mea- dimensional CCF by cross-correlating two template spectra (one surement errors due to line blending and/or the shifting of spec- for each star) with each target spectrum, which should reduce the tral features in or out of the analysed wavelength range due to or- problems caused by blending effects when the stellar spectral bital motion. The corrections are found by simulating observed lines are separated by less than their total broadening. As with spectra with known radial velocities and analysing them in the the onecor analysis, three types of template spectrum were in- same way as in the target spectra. vestigated and the broadened spectrum of HR 1855 was found to A set of simulated spectra were generated for each spectro- be the best. Radial velocities were measured by determining the scopic orbit, using HR 1855 (broadened) as a template, for the position of the relevant peak of the CCF by fitting a surface with observed phases and calculated velocities. A light ratio of 0.9 the IDL3 procedure min-curve-surf. They were accepted or was used and no additional simulated observational noise was rejected via visual inspection of the CCF, and orbits were calcu- added. We have derived these corrections for all four of our ve- lated using sbop as before. locity analysis methods because there was a priori no reason to The todcor algorithm can be used to analytically
 evaluate assume that the corrections would be negligible for any one of the light ratio between the two component stars B on the ba- them. The derived corrections were applied to the radial veloc- A sis of the relative strengths of their absorption lines (Zucker & ities from the target spectra and new spectroscopic orbits were Mazeh 1994). The mean and standard deviation of the light ratio calculated. calculated for each spectrum is given in Table 3 for each échelle The spectroscopic orbits without velocity corrections are order. These light ratios are meaningful because of the negligi- given in Table 4 and the corrected orbits are given in Table 5; ble difference in spectral type between the two components of it can be seen that in the case of DW Car the velocity correc- DW Car. tions generally amount to a compensation for the systematic er- ror caused by line blending. We have chosen the todcor analy- 3 http://www.rsinc.com/idl/ sis of the He I λ4471 line to illustrate these results and the nature 1082 J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae

Table 4. Spectroscopic orbits calculated for each component of DW Car, given for each of the considered He I lines and for each of the four radial velocity measurement techniques. KA and KB refer to the velocity semiamplitudes of the stars and Vγ,A and Vγ,B to the systemic velocities (all quantities in km s−1). Systemic velocities cannot be measured as directly using spectral disentangling, and no error estimates are available (see text for details).

Spectral line (Å) Échelle Gaussian fitting Cross-correlation Spectral order One-dimensional Two-dimensional disentangling He I 55 KA 259.0 ± 2.2 252.8 ± 1.5 254.5 ± 1.6 263 4026.189 KB 277.8 ± 1.5 265.5 ± 3.2 267.3 ± 3.3 277 Vγ,A −15.2 ± 2.0 −0.4 ± 1.5 −1.3 ± 1.6 Vγ,B −14.5 ± 1.3 −0.7 ± 3.1 −2.3 ± 3.2 He I 51 KA 261.4 ± 1.3 259.8 ± 1.8 257.2 ± 1.1 263 4387.928 KB 280.6 ± 2.0 280.2 ± 2.2 274.8 ± 2.0 279 Vγ,A −3.4 ± 0.9 −1.3 ± 1.7 −1.0 ± 1.0 Vγ,B −3.8 ± 1.5 −1.0 ± 2.2 −0.2 ± 1.9 He I 50 KA 261.9 ± 1.5 252.2 ± 2.0 255.4 ± 1.5 258 4471.681 KB 281.5 ± 2.2 274.6 ± 3.1 273.7 ± 2.7 277 Vγ,A −30.9 ± 1.4 +9.9 ± 2.0 +0.2 ± 1.5 Vγ,B −23.7 ± 2.0 +8.3 ± 3.1 +.55 ± 2.7 He I 45 KA 254.1 ± 2.0 251.6 ± 4.0 251.3 ± 3.6 255 4921.929 KB 276.3 ± 1.1 269.1 ± 3.0 270.6 ± 2.7 275 Vγ,A −11.3 ± 1.8 −3.4 ± 4.0 −4.4 ± 3.4 Vγ,B −13.0 ± 1.0 −9.9 ± 3.0 −9.8 ± 2.6 He I 33 KA 263 6678.149 KB 281

Table 5. As Table 4 but with velocity corrections applied (see text for details).

Spectral line (Å) Échelle Gaussian fitting Cross−correlation Spectral order One−dimensional Two−dimensional disentangling He I 55 KA 263.1 ± 2.1 262.8 ± 1.1 259.3 ± 1.6 263 4026.189 KB 282.2 ± 1.5 275.2 ± 2.0 275.0 ± 1.8 277 Vγ,A −0.4 ± 1.9 3.6 ± 1.1 2.0 ± 1.5 Vγ,B 0.4 ± 1.3 2.5 ± 1.9 0.8 ± 1.6 He I 51 KA 254.6 ± 1.0 251.4 ± 1.2 252.4 ± 1.1 263 4387.928 KB 272.1 ± 1.7 266.1 ± 2.0 267.6 ± 1.8 279 Vγ,A −17.5 ± 0.9 −18.9 ± 1.8 −18.6 ± 1.0 Vγ,B −15.5 ± 1.5 −19.2 ± 1.1 −18.1 ± 1.6 He I 50 KA 262.0 ± 1.2 257.6 ± 1.6 257.8 ± 1.4 258 4471.681 KB 281.6 ± 1.8 275.9 ± 2.6 273.9 ± 2.2 277 Vγ,A 0.0 ± 1.1 5.7 ± 2.6 1.3 ± 1.3 Vγ,B 3.0 ± 1.6 3.0 ± 1.6 1.4 ± 2.0 He I 45 KA 255.0 ± 2.0 252.4 ± 2.8 252.5 ± 2.1 256 4921.929 KB 276.9 ± 1.1 274.1 ± 1.9 272.2 ± 2.0 275 Vγ,A −6.2 ± 1.5 −7.4 ± 2.6 −6.9 ± 1.9 Vγ,B −8.4 ± 1.1 −9.2 ± 1.7 −10.5 ± 1.8 He I 33 KA 263 6678.149 KB 281 of the velocity corrections. The uncorrected radial velocities and one-dimensional CCFs suffer from line blending, it is not sur- best-fitting spectroscopic orbits are shown in Fig. 4, and the de- prising that this effect is still present in the todcor CCFs. rived corrections are plotted in Fig. 5. The corrected radial veloc- The velocity corrections required by 2gauss were a lot ities are shown in Fig. 6; note that the use of corrections allows smaller than those for onecor or todcor. This is understand- more velocities to be retained as reliable, improving the quality able because fitting a double Gaussian function explicitly com- of the resulting spectroscopic orbits. pensates for spectral line blending. The velocity corrections arise The two cross-correlation techniques require the largest ve- mainly due to slight departures in the shape of spectral lines from locity corrections, which is understandable because CCFs are ef- Gaussians. However, the corrections of 2gauss radial veloci- fectively smoothed by the rotational broadening both of DW Car ties have a larger effect on the resulting systemic velocities than and of the template spectrum, which was broadened to provide those for onecor and todcor. This is again due to slightly the closest match to the spectra of the components of DW Car. non-Gaussian spectral line shapes, and is particularly noticeable Whilst todcor was introduced to minimise the effects of spec- for the λ4471 line which has a blended forbidden component on tral line blending (Zucker & Mazeh 1994), the two-dimensional its short-wavelength side. CCF produced by the todcor algorithm is calculated from Velocity corrections to spectroscopic orbits calculated by the normal CCFs between the target and each template spec- disentangling were negligible, confirming in this case that the trum and between the two template spectra themselves. As the technique is an excellent way to measure spectroscopic orbits. J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae 1083

Fig. 4. Radial velocities and best-fitting spectroscopic orbit measured from analysing the λ4471 spectral line with todcor. No velocity cor- rections have been made. Circles represent velocities for star A and squares for star B. Rejected observations are shown using open sym- bols. They were rejected due to the appearance of their CCF, not the value of the derived velocities. See text for discussion.

Fig. 3. Variation of the quality of the fit for spectral disentangling the échelle order 55 spectra of DW Car. The upper panel is a surface plot of fit quality versus KA and KB. The lower panel is a contour plot of the same data with the six highest values of fit quality indicated by filled circles (largest for the best fit and smallest for the sixth-best fit). On the lower panel the area which a surface was fitted to is indicated by a dashed box and the highest point of that surface by a cross. Fig. 5. Corrections derived for the radial velocities measured from the λ4471 line using todcor. Values for star A are shown using filled cir- This suggests that spectral disentangling is not significantly af- cles and those for star B using asterisks. These corrections are an excel- fected by line blending. This conclusion was also reached by lent indicator of the effects of line blending. Hilditch (2005).

(including the size of the spectral range and whether the low- 3.6. Final Keplerian orbit est frequency components of the Fourier transforms are sup- The final velocity semiamplitudes for each radial velocity mea- pressed) can make a significant difference to the uncorrected ve- surement technique are given in Table 6 along with our adopted locity semiamplitudes. These differences normally vanish when values (velocity semiamplitudes from spectral disentangling and the velocity corrections are applied, illustrating how important systemic velocities from 2gauss). The velocity semiamplitudes the corrections can be. However, strongly blended lines continue are in good agreement with the values derived by Ferrer et al. to give unreliable velocities even when corrections are applied. −1 −1 (1985): KA = 261 ± 10 km s and KB = 278 ± 10 km s .Our Calculating corrections iteratively, using already corrected or- systemic velocities are in acceptable agreement with the −10 ± bits, improves this slightly, but cross-correlation cannot reliably 5and−16 ± 5kms−1 found by Ferrer et al. deal with spectra which are strongly affected by line blending. We have shown that velocity corrections are needed for Averaged spectroscopic orbits for each analysis technique onecor and 2gauss as well as for todcor, and that both cross- have been calculated by finding the mean and the standard er- correlation techniques seem to suffer from line blending even ror of the velocity-corrected orbital parameters in Table 5 for when these corrections have been included. We have also dis- different spectral lines (Table 6). It can be seen that the velocity covered that small differences in how the CCFs are calculated semiamplitudes calculated using onecor and todcor are lower 1084 J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae

Table 6. Keplerian spectroscopic orbital parameters for DW Car. The mean and standard error are given for each of the four radial velocity analysis techniques, and for the final results. All quantities are in units of km s−1).

KA KB VγA Vγ B 2gauss 258.7 ± 1.9 278.2 ± 2.0 −7.3 ± 3.7 −6.3 ± 4.0 onecor 256.0 ± 2.3 272.8 ± 2.0 −4.3 ± 4.9 −5.7 ± 4.6 todcor 255.5 ± 1.5 272.2 ± 1.4 −5.6 ± 4.2 −6.6 ± 4.1 Spec. dis. 260.4 ± 1.5 277.8 ± 0.9 Adopted 260.4 ± 1.5 277.8 ± 0.9 −7.3 ± 3.7 −6.3 ± 4.0

Table 7. Corrected radial velocities from a 2gauss analysis of the He I λ4471 spectral line. Note that these data are only a small part of those used to calculate the final spectroscopic orbits. The columns marked O−C give the observed minus calculated velocity residuals.

HJD − Weight Radial velocities O−C 2 400 000 star A star B A B 51 144.83060 1.0 257.4 –279.8 –3.0 –2.0 51 145.84399 0.0 25.0 13.2 28.5 8.8 51 146.86755 1.0 –256.0 284.5 5.2 4.5 51 147.86649 0.0 –1.4 48.9 34.9 9.4 51 148.80902 1.0 247.7 –278.5 –12.1 –1.4 51 149.87170 0.0 106.7 66.2 55.4 120.4 Fig. 6. Radial velocities and best-fitting spectroscopic orbit measured 51 150.83746 1.0 –262.2 276.5 0.5 –5.1 from a todcor analysis of the λ4471 spectral line with with velocity 51 152.73657 1.0 240.0 –275.1 –2.8 –16.1 corrections subtracted. Symbols are as in Fig. 4. The residuals of the fit 51 172.69544 2.0 256.2 –273.6 –1.2 1.0 are shown towards the bottom of the plot. Points were rejected on the 51 174.78088 2.0 –252.7 269.4 1.3 –2.8 basis of the apearance of their CCF, but taking into account the effects 51 176.80734 1.0 246.9 –245.4 9.3 8.0 of applying the corrections. 51 178.84953 1.0 –208.1 226.8 –1.6 5.4 51 179.87450 0.0 –192.6 216.3 –2.2 12.0 51 180.81349 1.0 230.1 –238.3 5.8 1.0 51 181.87490 1.0 193.7 –219.9 –2.4 –10.8 than those from 2gauss and disentangling. The main problem 51 182.86220 0.0 –184.0 209.9 –2.2 14.8 in determining these spectroscopic orbits is line blending, which 51 184.86864 0.0 167.6 –193.0 1.1 –15.6 will always act to reduce KA and KB. Therefore the techniques 51 187.87364 1.0 –220.4 233.8 –4.4 2.1 which are least affected by line blending should give higher KA 51 189.86682 1.0 215.1 –223.6 –0.5 6.2 and KB, implying that disentangling is best, closely followed by 51 190.84867 0.0 –207.3 118.5 –44.1 –56.6 2gauss, whereas the cross-correlation techniques, onecor and 51 191.86279 1.0 –229.4 238.5 –9.3 2.5 todcor, are significantly affected. 51 192.80846 1.0 199.0 –201.8 8.4 1.4 51 193.87953 1.0 232.0 –244.4 –2.1 5.2 We have adopted the averaged spectroscopic orbit (and stan- 51 197.85496 1.0 230.3 –254.2 0.7 –9.4 dard error) from disentangling as the best result, with the sys- 51 199.87004 2.0 –247.1 254.8 –3.5 –6.4 temic velocities from 2gauss (note that this is not the final 51 209.72395 1.0 164.3 –196.2 –1.9 –19.1 orbit: see below). The good agreement between this and the 51 382.51044 2.0 263.2 –270.3 1.9 8.6 average orbit for 2gauss shows that the technique of fitting dou- 51 386.50761 2.0 261.8 –273.0 0.5 5.9 ble Gaussians also performs very well in situations such as this. 51 563.86655 2.0 –221.5 251.4 6.1 7.4 It is important to note that the results we derived using spec- tral disentangling can only be fully trusted because the results from other techniques confirm that we have chosen the correct use of spectra from a dEB which exhibits a far richer set of spec- peaks in the surfaces of fit quality versus KA and KB (Fig. 3), al- tral lines. though in every case this peak was a global maximum. We have tabulated one set of radial velocities for the interested reader 3.7. Final spectroscopic orbit (Table 7). We chose the 2gauss analysis of the He I λ4471 line, as 2gauss seems to be more reliable than onecor or todcor, The spectroscopic orbits fitted so far have been purely Keplerian, and the λ4471 line velocities give a similar result to the adopted i.e. we have assumed that the velocities of the centres of light final orbit. of the stars are identical to those of the their centres of mass. It is also noticeable from Table 6 that the agreement between In close binaries, though, the effects of reflection and ellipsoidal different analyses of the same spectral line is greater than that stellar shape can cause the observed velocities to be significantly between the same analysis method used on different lines. This different to the centre-of-mass velocities. These effects are ac- suggests that the systematic errors of different techniques are curately modelled by the Wilson-Devinney code (see Sect. 4), smaller than the random errors, although it must be remembered which can be used to correct individual radial velocities using that the systematic errors may remain at full strength after the a given system geometry. However, our adopted velocity semi- results from several lines have been averaged. The systemic ve- amplitudes were calculated using spectral disentangling. With locities from different échelle orders have a poorer agreement this technique the spectroscopic orbit is calculated directly with- than expected; any further investigation of this will require the out the intermediate step of calculating radial velocities. We J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae 1085

Table 8. Final spectroscopic orbit after correction for non-Keplerian ef- uclsyn and was found to be 0.97 ± 0.01. The final light ra- fects. i and a are the orbital inclination and semimajor axis, respectively. tio, using the equivalent width measurements, is 0.84 ± 0.04 for the λ4026 line, which is close to the central wavelength of the Star A Star B Strömgren v passband, 4110 Å (Strömgren 1963). K (Keplerian) (km s−1) 260.4 ± 1.5 277.8 ± 0.9 K (corrected) (km s−1) 261.7 ± 1.6 279.3 ± 1.2 Mass ratio 0.9370 ± 0.0070 3.9. Spectroscopic helium abundance a sin i (R)14.192 ± 0.052 3 At the suggestion of the referee we have attempted to derive M sin i (M) 11.25 ± 0.12 10.54 ± 0.14 surface helium abundances for the components of DW Car. The high rotational velocities mean that this is difficult, but spectro- therefore cannot apply radial velocity corrections and so must scopic abundance analyses have been successfully performed on similar objects by Pavlovski & Hensberge (2005) and Pavlovski calculate corrections which can be applied to the velocity semi- ffi amplitudes themselves. et al. (2006). Our data contain insu cient information to sig- We have calculated spectroscopic orbits for the 2gauss ra- nificantly constrain the metal abundances of the components of DW Car. dial velocities, using the Wilson-Devinney code both with and atlas without the inclusion of non-Keplerian effects. This has been Model atmospheres were calculated using the 9 code done for échelle orders 55, 51, 50 and 45. The mean value (Kurucz 1979). Non-LTE atomic level populations were ob- tained for hydrogen and helium using the detail code and syn- and standard error of the resulting corrections to the veloc- surface ity semiamplitudes (in the sense K − K =∆K)are thetic spectra were constructed using the code (Butler true Keplerian ff ∆K =+1.3 ± 0.1kms−1 and ∆K =+1.5 ± 0.2kms−1.This & Giddings 1985). We adopted the e ective temperatures and A B light ratio for DW Car found below and calculated the residuals indicates that in this case fitting Keplerian orbits caused the ve- ff locity amplitudes (and so the masses) to be systematically under- of the best fit for several di erent helium abundances. estimated by a small but significant amount. This is the expected Formally, the best fits are found for a helium abundance result, because the light centres for each star are shifted towards equal to or slightly lower than the solar value (Fig. 7), but with the companion due to their mutual irradiation, causing the mea- a large uncertainty. Varying the metal abundance within reason- ff sured radial velocities to be underestimates of the motion of the able limits has no significant e ect. However, whilst the fit of centre of mass. the lines at 4388 Å and 4471 Å are excellent, the 4026 Å line is The final spectroscopic orbit, with the non-Keplerian correc- too shallow in the synthetic spectrum whilst the 4922 Å line is tion applied, is given in Table 8. The resulting minimum masses too deep. The investigation of these effects is beyond the scope 3 3 are MA sin i = 11.25 ± 0.12 M and MB sin i = 10.54 ± of this work. We conclude that the spectra are consistent with 0.14 M where i is the orbital inclination. The corrections to a solar or subsolar helium abundance but not with an abundance the systemic velocities are negligible and have not been applied. signficantly greater than solar. The rotational velocities were also optimised when fitting the theoretical spectra to the disentangled spectra, and the best- 3.8. Spectroscopic light ratio −1 fitting values are VA sin i = 182 ± 3kms and VB sin i = ± −1 The light ratio of the two stars is poorly defined by the light 177 3kms . curves of the system (see below) so we have derived a spectro- scopic light ratio to help constrain the light curve solutions. The todcor algorithm was used to analytically determine the light 4. Light curve analysis ratio for each spectral line analysed (see Sect. 3.3 and Table 3), yielding an average value and standard deviation of B = 0.873 ± The light curves of DW Car contain 518 photoelectric observa- A 0.042. The wavelength dependence of the light ratio is negligi- tions in each of the uvby passbands (Paper I). The two eclipses are relatively long in phase units and are both about 0.7 mag ble between the Balmer jump and 5500 Å (see Sect. 4.1). This is deep, indicating that the system is composed of two similar and a reliable equivalent width ratio measurement because the com- close stars. We have used the 2003 version (Van Hamme & ponents of DW Car have very similar spectral characteristics, Wilson 2003) of the Wilson-Devinney code (Wilson & Devinney and the same template spectrum was used for both stars. 1971; Wilson 1979, 1993), hereafter called wd2003, to model The equivalent widths of the He I λ4026 and λ4471 lines for the light curves, as Roche geometry is needed to adequately rep- both stars were also measured directly using several spectra with resent the tidal deformation of the two stars. wd2003 optionally large velocity separations and high signal to noise ratios. The uses the predictions of the atlas9 model atmospheres (Kurucz component lines are still slightly blended so we have used the ff iraf4 splot 1993) to connect the flux ratios of the two stars in di erent deblending feature of the task to measure the line passbands. strengths. The mean values and standard deviations are 0.87 ± 0.03 and 0.89 ± 0.03 for the λ4026 and λ4471 lines respectively, Initial investigations showed that the orbital eccentricity is in good agreement with the todcor results. negligible, so we assumed that the orbit is circular. The rotational The He I lines are intrinsically slightly stronger in star B velocities of the stars were fixed at the synchronous values as these are consistent with the spectral line widths. The bolomet- (T ff = 26 500 K) than in star A (T ff = 27 900 K), so the equiva- e e ric albedo coefficients and gravity brightening exponents were lent width ratios are slightly different to light ratios. A correction was derived using intrinsic equivalent width ratios taken from fixed at 1.0, which is the expected value for stars with radia- tive envelopes (Claret 1998). The mass ratio was fixed at the 4 iraf is distributed by the National Optical Astronomical spectroscopic result of 0.937 and we chose to model reflection Observatories, which are operated by the Association of Universities for using the faster and simpler of the two alternatives implemented Research in Astronomy, Inc. under contract with the National Science in wd2003. We have used grid sizes of N = 20 for each star; Foundation. solutions with finer grids gave negligibly different results. 1086 J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae

Fig. 7. Plot of the normalised disentangled profiles of four helium lines for each star compared to theoretical spectra calculated for helium abun- dance Y = 0.28 and the effective temperatures and light ratio found in our analysis. The left-hand panels are for star A and the right-hand panels for star B. The observed spectra are plotted using points and the theoretical spectra using solid lines.

We expected that there would be a small amount of contam- constrain the solution of the v band light curve. This was done inating “third” light in the light curves of DW Car, as its Be star by fixing the potential of star A at values which gave the desired nature indicates that there is substantial circumbinary material. light ratio and plus or minus its uncertainty. These values for po- Third light was therefore included as a fitting parameter in the tential were then fixed for the uby light curve solutions. The po- light curve analysis, but the optimised values were all consis- tential of star B, the orbital inclination, the phase shift between tent with, and marginally lower than, zero. We therefore have primary midminimum and phase zero, and the light contribu- assumed that third light is zero. A small amount of third light tions for the two stars were optimised for each light curve. The (up to about 5%, which is an order of magnitude greater than linear terms in the square-root limb darkening law were also op- the best-fitting values) will have a negligible effect on the light timised. The effective temperatures of the two stars were fixed curve solutions. at nominal values (27 500 and 26 750 K), but these do not affect Limb darkening coefficients were initially set to values found the final solution because we are fitting for the light contribu- by bilinear interpolation in effective temperature and surface tions of both stars directly. Figure 8 shows the light curve fits gravity from the coefficients tabulated by Van Hamme (1993), and residuals. but in our final solutions we have always optimised these val- Table 9 contains the values of the optimised parameters, as ues to avoid the theoretical dependence caused by using coef- well as the standard errors calculated by wd2003, for each light ficients calculated from stellar atmosphere models. The choice curve and for each of the three potentials of star A (4.101, 4.132 of linear, logarithmic or square-root law has a negligible effect and 4.157) which are required to reproduce the light ratios 0.80, on the solutions so we have chosen to use the latter, as this pro- 0.84 and 0.88 for the v light curve. We have adopted the results vides the best fit to model atmosphere predictions for hot stars for a star A potential of 4.132 as the final values. The uncertain- (Van Hamme 1993). Fixing the limb darkening coefficients at ei- ties in these light curve parameter values have been calculated ther the Van Hamme (1993) or Claret (2000) theoretical values by adding in quadrature an uncertainty from the variation for causes a negligible change in the final solution – of the order different potentials to the mean standard error for each param- of 0.1% in the stellar radii. The optimised values are generally eter. Additional uncertainties due to the treatment of third light in good agreement with the theoretically predicted coefficients. and limb darkening are negligible. The radii found by fitting the light curves with wd2003 are strongly correlated with each other and with the light ratio, re- 4.1. Strömgren indices for each star sulting in one of many different light curve solutions (i.e., local minima in parameter space) being found depending on the ini- Separate Strömgren photometric indices can be derived for each tial fitting estimates. The residuals of the fit are approximately star from the indices of the system and the light ratios in the constant for ratios of the radii between 0.8 and 1.2. We have uvby passbands. This requires a solution of the light curve with therefore used the light ratio of 0.84 ± 0.04 found in Sect. 3.8 to a consistent geometry, so we have derived the flux ratios and J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae 1087

Fig. 8. Light curves and best wd2003 fit with the potential of star A fixed at 4.1706 (upper panel) and the residuals of the fit (lower panel)shown with an expanded scale in magnitudes.

corresponding Strömgren indices (and uncertainties) by fitting reliable in this case as the effective temperatures of the two stars the light curves with the potential of star A fixed at the values are very similar. Using this option, we have obtained an effec- found above. For each fixed potential, we have calculated the tive temperature difference of TeffA − TeffB = 1380 ± 420 K for Strömgren indices for each star from the indices of the combined the stars in DW Car, where the error has been found by adding system and the light ratios in the four light curves (Table 9). in quadrature the uncertainty caused by varying the potential of The uncertainties in Table 9 have been calculated by adding in star A as before, and the formal error of the temperature differ- quadrature all the relevant uncertainties, including those result- ence calculated from the covariance matrix by wd2003. Formal ing from the range of possible values of the potential of star A. errors can be underestimates of the real uncertainties of light They are somewhat lower than they would be if we had simply curve parameters (Popper 1984; Maceroni & Rucinski 1997; used the final light ratio errors for each light curve, because the Southworth et al. 2005a). light ratios are correlated with each other (i.e., a change in the potential of star A causes absolute changes in the light ratios significantly larger than they change relative to each other). 5. Absolute dimensions The uvby light curves have also been solved simultaneously The masses, radii and surface gravities of the two components to derive an effective temperature difference for the two com- of DW Car have been calculated from the results of the spectro- ponent stars. In this case, the light ratios in different passbands scopic and photometric analyses. Using careful error propaga- must be connected, which can be achieved either using Planck tion, we find that the masses and radii are known to accuracies of law or by using the predictions of theoretical model atmo- about 1% (Table 11). DW Car therefore joins the (very short) list spheres. The Planck law is not a good option for DW Car because of well-studied high-mass unevolved dEBs, alongside QX Car the Balmer jump causes the stars’ radiative properties to be quite (Andersen et al. 1983) and V578 Mon (Hensberge et al. 2000). different from black bodies. wd2003 optionally uses predic- We have derived the effective temperatures of the two stars tions from Kurucz atlas9 model atmospheres, which should be from the Strömgren indices found in Sect. 4.1 using the 1088 J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae

Table 9. wd2003 light curve solutions for DW Car. A light ratio was used to constrain the solution for the v light curve, and the resulting potential of star A was fixed for the other light curves (see text for further details). The quoted errors (except for the adopted values in the final column) are the standard errors calculated by wd2003.

Parameter Potential vybuAdopted of star A ±±±±values Light ratio 4.101 0.800 0.796 ± 0.012 0.797 ± 0.009 0.769 ± 0.034 4.132 0.840 0.840 ± 0.014 0.840 ± 0.010 0.813 ± 0.034 4.157 0.880 0.879 ± 0.013 0.881 ± 0.010 0.858 ± 0.032 Potential of 4.101 4.225 ± 0.007 4.236 ± 0.009 4.228 ± 0.007 4.248 ± 0.011 star B 4.132 4.192 ± 0.007 4.200 ± 0.009 4.193 ± 0.007 4.211 ± 0.010 4.157 4.157 ± 0.007 4.167 ± 0.008 4.158 ± 0.007 4.172 ± 0.009 Fractional radius 4.101 0.3237 0.3236 0.3237 0.3237 0.3203 of star A 4.132 0.3203 0.3203 0.3203 0.3203 ± 0.0030 4.157 0.3176 0.3175 0.3175 0.3175 Fractional radius 4.101 0.2993 ± 0.0007 0.2982 ± 0.0009 0.2990 ± 0.0007 0.2970 ± 0.0011 0.3019 of star B 4.132 0.3026 ± 0.0007 0.3018 ± 0.0009 0.3025 ± 0.0007 0.3007 ± 0.0010 ± 0.0037 4.157 0.3063 ± 0.0007 0.3052 ± 0.0009 0.3061 ± 0.0007 0.3048 ± 0.0010 Orbital inclination 4.101 85.76 ± 0.07 86.00 ± 0.09 85.96 ± 0.07 86.13 ± 0.08 85.72 (degrees) 4.132 85.61 ± 0.07 85.77 ± 0.04 85.75 ± 0.04 85.74 ± 0.06 ± 0.28 4.157 85.63 ± 0.07 85.73 ± 0.08 85.70 ± 0.07 85.83 ± 0.09 Phase shift of primary 4.101 0.06 ± 0.04 –0.02 ± 0.03 –0.02 ± 0.05 0.03 ± 0.05 midminimum (10−3) 4.132 0.08 ± 0.05 –0.03 ± 0.07 –0.01 ± 0.05 0.01 ± 0.05 4.157 0.06 ± 0.04 –0.04 ± 0.04 0.00 ± 0.05 0.04 ± 0.04 Linear LD coefficient 4.101 –0.124 ± 0.031 –0.132 ± 0.037 –0.156 ± 0.029 –0.056 ± 0.038 of star A (using the 4.132 –0.084 ± 0.031 –0.091 ± 0.038 –0.113 ± 0.030 –0.023 ± 0.038 square-root law) 4.157 –0.125 ± 0.036 –0.114 ± 0.043 –0.137 ± 0.036 –0.039 ± 0.045 Linear LD coefficient 4.101 –0.093 ± 0.042 –0.087 ± 0.050 –0.117 ± 0.041 –0.069 ± 0.058 of star B (using the 4.132 –0.095 ± 0.042 –0.068 ± 0.052 –0.109 ± 0.042 –0.017 ± 0.057 square-root law) 4.157 –0.145 ± 0.043 –0.114 ± 0.052 –0.149 ± 0.042 –0.065 ± 0.058 Size of residuals 4.101 5.21 6.20 4.97 7.15 σrms (mmag) 4.132 5.16 6.23 5.00 7.04 4.157 5.39 6.23 5.02 7.04

calibration of Moon & Dworetsky (1985) contained in in their (b − y) colours. We will adopt Eb−y = 0.18 ± 0.02 for the code uvbybeta (Moon 1985). Using the individual DW Car, where the quoted error includes a contribution from the Strömgren indices for each star, we find TeffA = 27 800 ± 1100 estimated uncertainty in the calibration. and TeffB = 26 500 ± 1300 K, where the error does not in- clude any uncertainty in the calibration and its main contribu- 5.1. The distance to DW Car tion comes from the uncertainty in the m1 indices. However, the equivalent effective temperature of the system is much bet- Now we have the absolute dimensions and effective tempera- = ± ter defined: TeffA+B 27 230 390 K, and the temperature tures of the two stars we can calculate the distance to the sys- difference is also well defined from the light curve analysis: tem. Distances are in general more accurate and precise at in- − = ± TeffA TeffB 1380 420 K (formal error). Using these quan- frared wavelengths (Southworth et al. 2005a), but we are lacking tities, and including an estimated uncertainty in the photometric reliable apparent magnitudes in the JHKL passbands. These calibration, we find temperatures of 27 900 ± 1000 and 26 500 ± would be available from 2MASS, but are too faint because they 1000 for the two stars. The calibrations of Davis & Shobbrook were taken during secondary eclipse. JHKL photometry is avail- (1977) and Napiwotzki et al. (1993) give temperatures in good able from Tapia et al. (1988) but seems to be too bright com- agreement with these values. pared to the optical colours of DW Car. The emission-line na- The equatorial rotational velocities quoted for each star were ture of DW Car suggests that there may be a circumbinary disc. ff found by comparing the disentangled spectra of the stars to the- This may a ect the infrared colours of the system, resulting in oretical spectra in order to determine the surface helium abun- brighter apparent magnitudes. We are therefore unable to use in- dance (Sect. 3.9). They are slightly greater than the synchronous frared magnitudes of DW Car to measure its distance with any value despite the short orbital period of DW Car. This indicates confidence. that it is a young system in which the rotational and orbital ve- We have used the V band magnitude and flux ratio of DW Car locities have not yet been equalised by tidal effects. (Table 10) to determine the distance to each star and to the sys- tem using several sources of bolometric corrections. We used The reddening of the system has been found using the the jktabsdim code5, which calculates distances using sev- uvbybeta − code, which implements the (b y)0 vs. c0 rela- eral different sources of bolometric corrections (Southworth tion by Crawford (1978). The reddening for the system in- et al. 2005a). The available sets of bolometric corrections are = ± dices and for the individual stars is Eb−y A+B 0.180 0.007, calculated using, in general, different radiative parameters for the = ± = ± Eb−y A 0.177 0.054 and Eb−y B 0.182 0.064, where Sun (Bessell et al. 1998). It is important to adopt the appropriate the quoted errors come only from varying the indices within their uncertainties. The main contribution to the reddening un- 5 This can be obtained from certainties for the individual stars comes from the uncertainty http://www.astro.keele.ac.uk/∼jkt/codes.html J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae 1089

Table 10. Light ratios of DW Car for a consistent geometry, with the Table 11. The physical parameters of the component stars of DW Car resulting Strömgren indices. Note that the quoted flux ratio errors were derived from the results of the spectroscopic and photometric analyses. not used in calculating the errors of the indices of each star. Veq denotes the observed equatorial rotational velocities of the stars. 26 † Calculated assuming L = 3.826 × 10 WandMbol = 4.75 y b v u (Zombeck 1990). Flux ratios 0.840 0.840 0.840 0.813 ± 0.046 0.044 0.040 0.056 Parameter Star A Star B Vb− y m1 c1 Mass (M) 11.34 ± 0.12 10.63 ± 0.14 System 9.675 0.067 0.046 −0.017 Radius (R) 4.558 ± 0.045 4.297 ± 0.055 ± 0.005 0.005 0.008 0.008 log (g/cm s−3) 4.175 ± 0.008 4.198 ± 0.011 − −1 Star A 10.307 0.067 0.046 0.034 Veq (km s ) 183 ± 3 178 ± 3 ± −1 0.026 0.011 0.016 0.023 Vsynch (km s ) 173.6 ± 1.7 163.7 ± 2.1 Star B 10.563 0.067 0.046 0.004 log Teff (K) 4.446 ± 0.016 4.423 ± 0.016 ± 0.032 0.014 0.020 0.030 log (L/L)† 4.055 ± 0.063 3.915 ± 0.067 Mbol†−5.39 ± 0.16 −5.04 ± 0.17 values when using bolometric corrections, and improper results can be obtained if this is not done (see discussion in Southworth In dEBs such as DW Car, which are composed of two simi- et al. 2005c). lar stars, the effective temperatures can be very similar and the Adopting the bolometric correction formula of Flower errorbars often overlap. In this case, the temperature difference (1996), we find that the individual distances of the two stars are suggested by the individual temperatures (Table 11) is 1400 ± in acceptable agreement (2877 and 2742 pc), and the overall dis- 1420 K, whereas results from the light curve analysis give tance measurement to the system is 2810 ± 160 pc, where the a much more precise temperature difference of 1380 ± 420 K. main contributor to the uncertainty comes from the reddening. If This can be represented on a mass-temperature plot by giving we use the theoretically-derived V band bolometric corrections full-size errorbars to one star (here star A) and only relative er- of Bessell et al. (1998) or Girardi et al. (2002) we find distances rorbars to the other star (B). to DW Car of 2710 ± 140 pc and 2720 ± 140 pc, respectively. We have chosen to compare the properties of DW Car with The agreement between the different distance measurements is theoretical predictions from the Claret evolutionary models acceptable given the uncertainties, but not as good as we were (Claret 1995, 1997; Claret & Giménez 1995) which are avail- expecting. The components of DW Car are hot stars, and trans- able for several fractional helium abundances, Y, for each frac- forming their visual absolute magnitudes into bolometric ones tional metal abundance, Z. These models use the opacities of requires corrections which are large and relatively uncertain. Iglesias et al. (1992) and incorporate moderate convective core An improved distance estimate could be found by studying the overshooting (with αOV = 0.20) and mass loss, and are given for ultraviolet to optical spectral energy distribution of the system scaled-solar metal abundances. (see Guinan et al. 1998; and Fitzpatrick & Massa 1999). Isochrones for the best-fitting ages are plotted in Fig. 9 for Using the RI photometry of DW Car from Carraro & Patat metal abundances of Z = 0.004, 0.01 and 0.02. A subsolar metal (2001) and bolometric corrections from Girardi et al. (2002) abundance, Z = 0.01, provides an excellent match to the masses ± ± gives distances of 2890 130 pc and 2710 110 pc, for R and radii of DW Car, but predictions for the other chemical com- and I respectively, where the decreasing importance of the un- positions are also in reasonable agreement here. However, the certainty in reddening towards longer wavelengths can be seen. standard helium abundances in the Claret models (dashed lines) ± However, we will adopt 2810 160 pc as our final distance, give rise to predicted temperatures slightly too cool for Z = 0.01 found from the V band photometry and the Flower bolometric and 0.02, although good agreement is found for Z = 0.004. correction formula, because this is an empirical measurement. A slightly enhanced helium abundance gives a better fit. All The corresponding distance modulus is 12.24 ± 0.12 mag. models predict shallower mass-Teff slope for DW Car, although they agree within the uncertainties. Other sets of model predictions (Schaller et al. 1992; Pols 6. Comparison with theoretical models et al. 1998) were investigated and similar results were found. Graphical comparisons between the predictions of theoretical “Classical” models (without convective core overshooting; Pols stellar models and the physical properties of a dEB provide et al. 1998), gave predictions which were negligibly different a clear and straightforward interpretation of the agreement or from their equivalent overshooting models, as expected for such disagreement between theory and observation. The most impor- young stars. tant plot is of mass versus radius, as these are the two quan- Whilst we found the surface helium abundance of DW Car to tities which have been directly measured for the dEB (i.e., be approximately solar (Sect. 3.9), the model comparison sug- without using any calibrations). A second plot can be made gests an overall abundance slightly greater than solar. Note that because we have measured both the radii and effective tem- these two statements are not mutually inconsistent because they peratures of two stars with known masses. Chosing the ordi- refer to different parts of the stars. However, neither abundance nate quantity to be mass is again the best approach because the measurement is very accurate. We conclude that the best match masses of the stars have been directly measured and because between stellar models and the properties of DW Car is found model predictions are calculated for specific (initial) masses. for a metal abundance of Z = 0.01, a helium abundance of The abscissal quantity could be either effective temperature or Y = 0.26 (roughly solar), and an age of 6 Myr. However, these luminosity (or, equivalently, absolute bolometric magnitude); we three quantities are all quite uncertain and have a dependence on prefer the former because this quantity has been measured more theoretical models. The age and chemical composition we infer directly than luminosity. for DW Car are relatively imprecise because its two component 1090 J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae

Fig. 9. Comparison between the properties of DW Car and the predictions of the Claret stellar evolutionary models. Panels a) and b) are a simul- taneous match for metal abundance Z = 0.004, panels c) and d) likewise for Z = 0.01 and panels e) and f) for Z = 0.02. The model helium abundances and interpolated ages are given in the legend of each left-hand panel. The errorbars plotted for the effective temperature of star B are relative to the effective temperature of star A and do not represent the overall uncertainty. stars have similar properties. Much more precise conclusions are 7. DW Car and Collinder 228 possible for dEBs with smaller mass ratios (e.g., Southworth et al. 2004b). DW Car is a photometric member of Cr 228 and has a sky position close to the cluster centre. However, the systemic J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae 1091 velocities of Cr 228 quoted by Rastorguev et al. (1999) and spectral disentangling (Simon & Sturm 1994), although this pro- Levato et al. (1990), −12 km s−1 and −13.5kms−1 respec- cedure is not straightforward due to a profusion of local maxima tively, disagree slightly with our measured systemic velocity in the surface of the quality of the fit in parameter space. for DW Car (Table 6). However, as noted in Sect. 3.7, the sys- We have found that the radial velocities derived from temic velocities from the different échelle orders are not in full Gaussian fitting and from cross-correlation require substantial agreement considering their formal errors. Ferrer et al. (1985) corrections for the effects of line blending. These corrections found systemic velocities for the components of DW Car in good were derived from simulated composite spectra, constructed us- agreement with the cluster velocity. As our measured distance ing a broadened observed template spectrum, for radial velocity and reddening are also appropriate for cluster membership, it offsets appropriate for each observed spectrum. The corrections seems most probable that DW Car does belong to Cr 228. We were the best-behaved for Gaussian fitting, despite the obvious can therefore derive the important astrophysical properties for shortcoming of this technique that helium lines do not have Cr 228 from those for DW Car. strictly Gaussian shapes. However, the corrections for cross- Thedistancewederive,(m − M)0 = 12.24 ± 0.12 mag, is correlation do not seem to have been completely successful, in in good agreement with most of the measurements in the lit- that the resulting spectroscopic orbits have systematically lower erature, including that of Thé et al. (1980) for a normal red- ampitudes that those derived from the other two methods. The dening law, but not including those of Carraro & Patat (2001) line blending problem is more severe for cross-correlation, be- and Tapia et al. (1988). The disagreement with Tapia et al. re- cause both the object and the template spectra are broadened sults from the very large interstellar extinction value they find (using an unbroadened template spectrum causes excessive ran- using infrared photometry. Our distance is the most accurate of dom errors in the resulting radial velocities). Gaussian fitting, on the measurements, demonstrating that dEBs are excellent dis- the other hand, explicitly accounts for the presence of blending tance indicators. Our reddening value for DW Car, equivalent lines from the other star, so gives results which are reliable. The to EB−V = 0.25 ± 0.03 mag (Crawford 1978), is in reasonable cross-correlation results could probably be improved by fitting agreement with most literature values. single-lined CCFs to the double-lined CCFs (see Hill 1993). The The age of Cr 228 is less well known. Determinations on the broadening-function approach advocated by Rucinski (2002) is basis of photometry alone are tricky due to the sparse nature of also a useful alternative. The poor performance of todcor in the cluster, difficulties with measuring reddening, and because dealing with line blending arises because it presents no signifi- UBV photometry is a poor temperature indicator for hot stars cant advantage over traditional cross-correlation when the com- (e.g., Massey & Johnson 1993). Massey et al. (2001) used clas- ponent stars have very similar spectral characteristics (P. F. L. sification spectroscopy to measure ages for many stars in Cr 228 Maxted, private communication). and found an age of about 2 Myr, in good agreement with our The orbits measured from spectral disentangling were the estimate of about 5 Myr from comparison with the Claret evolu- most internally consistent, and did not require correcting for tionary models. blending. Disentangling should be innately more appropriate to The chemical composition of Cr 228 has not previously been this situation because it assumes nothing about the spectral char- studied. We find a metal abundance of Z ≈ 0.01 and an approx- acteristics of the stars, and uses (almost) every observed spec- imately solar helium abundance from the properties of DW Car. trum in a simultaneous least-squares fit. However, it suffers from A spectroscopic abundance study of some of the more slowly- the presence of many local maxima in the surface of quality rotating members of Cr 228 would be very useful in specifying of the fit to the observations. The best approach for measuring the chemical composition of the cluster and of DW Car, which spectroscopic orbits in the presence of strong line blending is would provide an additional constraint when comparing model therefore to use spectral disentangling, checked and corrob- predictions to the properties of this system. orated by the results of fitting double Gaussian functions. Conducting such analyses for many lines (helped if the spec- tra have the wide wavelength coverage achievable using échelle 8. Summary spectrographs) is very useful as it allows the overall errors to be DW Carinae is a young, high-mass detached eclipsing binary both reduced and robustly estimated. system with a short orbital period. It is particularly interesting The light curves of DW Car were modelled using the because it is unevolved, shows Be star emission-line character- Wilson-Devinney code. The ratio of the radii of the stars is istics, and is a member of the young open cluster Collinder 228 intrinsically poorly determined by the light curves due to cor- in the Carina complex. relations between various photometric parameters, as is often The spectra of DW Car contain strong hydrogen lines, with found for systems with deep but not total eclipses. This degen- some superimposed Be-type emission from circumbinary mate- eracy was broken by constraining the light curve solution using rial, but there are very few other noticable absorption lines. As a spectroscopically-derived light ratio. We also fitted directly for a result of this, we were only able to measure radial velocities the limb darkening coefficients of the stars in order to avoid a de- from four or five helium lines. The high rotational velocities of pendence on theoretically-derived coefficients. The adoption of the stars mean that other lines are not detectable, or that small different limb darkening laws (linear, square-root and logarith- changes in continuum placement can cause a large change in mic) makes a negligible difference to the result, and the fitted the derived radial velocities. Due to the high rotational veloci- coefficients are also in good agreement with the predictions from ties, we measured radial velocities using several ways to inves- model atmosphere analyses. tigate which are the most reliable in the presence of strong line The masses and radii of the components of DW Car, calcu- blending. lated from the results of the spectroscopic and photometric anal- Individual radial velocities were obtained by fitting dou- yses, are known to accuracies of about 1%, so this system joins ble Gaussians, by cross-correlation, and by the two-dimensional the short list of well-studied dEBs containing components with cross-correlation algorithm todcor (Zucker & Mazeh 1994) us- masses above 10 M.Theeffective temperatures, and interstel- ing the spectrum of a standard star as a template. Spectroscopic lar extinction were derived using photometric calibrations and orbits were also obtained directly (by least-squares fitting) using the individual Strömgren indices, which were calculated for the 1092 J. Southworth and J. V. Clausen: Absolute dimensions of the eclipsing binary DW Carinae two stars from the combined indices and the light ratios found material from the surrounding environment. The unevolved sta- during the light curve analysis. tus of DW Car is also consistent with the field population of The properties of DW Car can be used to infer the proper- high-mass Be stars (Zorec et al. 2005). ties of its parent cluster, Cr 228. The distance modulus of the dEB, calculated using the empirical bolometric corrections of Acknowledgements. A. Kaufer, O. Stahl, S. Tubbesing, and B. Wolf kindly Flower (1996), is 12.24 ± 0.12 mag, which is the most accu- observed nearly all the spectra of DW Car during the two periods of rate measurement of Cr 228’s distance yet. A comparison be- Heidelberg/Copenhagen guaranteed time at FEROS in 1998 and 1999. We thank tween the properties of DW Car and predictions from the theoret- H. Hensberge for making his improved version of the MIDAS FEROS package ical evolutionary models of Claret (1995) was also undertaken. available, and both him and L. Freyhammer for valuable advice and help during the reduction of the spectra. We are grateful to E. Sturm for providing his orig- The mass-radius and mass-Teff diagrams are the best choice for inal disentangling code, and to him and J. D. Pritchard for modifying it for use such a comparison, because they rely on the properties which are on Linux/Unix computer systems. We would also like to thank P. F. L. Maxted most directly measured and are most easily interpreted. A good and D. J. Lennon for useful discussions, P. F. L. Maxted for implementing our todcor match to the properties of the dEB is found for an age of roughly version of , and the (anonymous) referee for providing a timely and ≈ useful report. J.S. acknowledges financial support from the Instrument Centre 6 Myr, a metal abundance of Z 0.01, and a solar helium abun- for Danish Astrophysics (IDA) in the form of a postdoctoral grant. The project dance. We have modelled the helium lines of both components “Stellar structure and evolution – new challenges from ground and space ob- of DW Car, using non-LTE calculations, to find a roughly so- servations”, carried out at Aarhus University and Copenhagen University, is sup- lar or slightly subsolar helium abundance which is in acceptable ported by the Danish National Science Research Council. Additional support was agreement with that inferred from comparison with the theoreti- received from the (former) Danish Board for Astronomical Research. The fol- lowing internet-based resources were used in research for this paper: the NASA cal models. The age is in good agreement with literature values Astrophysics Data System; the SIMBAD database operated at CDS, Strasbourg, for Cr 228, and the inferred metal and helium abundances are France; the VizieR service operated at CDS, Strasbourg, France; and the arχiv (to our knowledge) the first measurements for this cluster, il- scientific paper preprint service operated by Cornell University. This publica- lustrating the usefulness of detached eclipsing binaries for the tion makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing investigation of stellar clusters. and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. 8.1. 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